Although it is natural to model the observed counts as multinomial distributed, it may not be obvious why we use Gaussian latent variables for the latent space. A conjugate distribution such as Dirichlet may seem more natural than the Gaussian distribution, which is not conjugate to Multinomial. However, the components of the Dirichlet distribution are nearly independent [1]}, hence it is non-trivial to capture the correlations between the hidden components. On the other hand, the Multivariate normal distribution has a covariance parameter which specifically captures the correlation between the hidden components. This is useful for modeling the correlation between multiple datasets. Similar model assumptions are also adopted in topic models [1]}, [3]}, categorical PCA [4]}, and Gaussian process classification [5]}.
Note that, although the communities are dependent through the latent variables, the experimental conditions are modeled as independent. Hence, there is no coupling between the experimental conditions and thus we fit independent models for each condition.
